ICM 2026 Special Section Lecture - Rachel Greenfeld

Value of the Information & Strength of the Argument

The lecture provides a rich overview of current research on translational tilings, presenting both classical results and novel contributions. The argumentation is methodical and rigorous: the speaker introduces precise definitions, explains the logical dependencies, and carefully contrasts known theorems with open problems. The progression from simple examples to powerful counterexamples is clear, demonstrating the depth of the subject and the elegance of the mathematical tools involved. The value lies in both the synthesis of prior work and the presentation of original results, making it an excellent reference for mathematicians and advanced students.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is well-sourced, citing foundational works by Berger, Golomb, Lagarias, Wang, and others, as well as recent contributions by the speaker and collaborators. The sources are used accurately to support each stage of the argument, and the state of the art is correctly represented. The title precisely matches the content, which explores the evolution from Wang dominoes to p-adic Sudoku. There is no evident discrepancy between the title and the actual material presented. No comments were analyzed; the lecture has very low visibility on the platform, but this does not affect the accuracy or rigor of the content.

The title accurately reflects the content: a lecture on translational tilings, from Wang dominoes to p-adic Sudoku, as delivered by Rachel Greenfeld at ICM 2026.

10/10

Lecture by a leading researcher at an ICM special section, presenting peer-reviewed results and rigorous mathematical arguments. References classical and recent literature in a clear, structured manner.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction to tilings with examples (beehive, Penrose, Spectre) and definition of translational tilings.
• 04:00Wang dominoes and the decidability problem; Berger's undecidability proof and construction of aperiodic tilings.
• 08:00Formal definition of translational tilings, the tiling space, and basic properties (translation invariance, reflection invariance, uniform density).
• 12:00Tools for studying tilings: graph theory (Keller graphs), Fourier analysis, cyclotomic methods, and dynamical systems.
• 18:00Periodic tiling conjecture (PTC) and supporting results in Z, R, Z², and virtually Z² groups.
• 22:00Dilation lemma and reversible structure theorem; proof of PTC in Z² using polynomial analysis.
• 26:00Counterexample to the periodic coloring conjecture via p-adic Sudoku; construction of aperiodic translational tilings in virtually Z² spaces.
• 31:00Undecidability of translational monotilings in virtually Z² spaces by encoding decorated p-adic Sudoku problems.
• 35:00Conclusion: wild behavior in high dimensions vs. structure in low dimensions; open questions for intermediate dimensions.

• Berger, R. (1966). The undecidability of the domino problem. — Mentioned as establishing the undecidability of the Wang domino problem and constructing a large aperiodic set of tiles.
• Golomb, S. W. (1970) – relation between Wang tilings and translational tilings of Z². — Cited to show that Wang tilings correspond to translational tilings with multiple tiles.
• Lagarias, J.C., & Wang, Y. (1996) – proof of the periodic tiling conjecture in R. — Mentioned as solving the PTC in the real line using Fourier analysis.
• Greenfeld, R., & Tao, T. (2020) – alternative proof in Z². — Described as a quantitative proof using combinatorial and commutative algebra techniques.
• Greenfeld, R., et al. – recent counterexample to periodic tiling conjecture in virtually Z² spaces. — Central result presented in the lecture, disproving the conjecture via p-adic Sudoku and tiling language.

• Penrose tiling – classic example of aperiodic tiling by translation and rotation. — Mentioned in the introduction as a well-known aperiodic tiling, illustrating the contrast with translational tilings.
• Periodic tiling conjecture – statement and history. — Some background on the conjecture that the lecture addresses, including positive results and recent counterexamples.